THE ABSOLUTE GALOIS GROUP OF SUBFIELDS OF THE FIELD OF TOTALLY S-ADIC NUMBERS

For a finite set S of local primes of a countable Hilbertian field K and for sigma(1),...,sigma(e) is an element of Gal(K) we denote the field of totally S-adic numbers by K-tot,(S), the fixed field of sigma(1),...,sigma(e) in K-tot,(S) by K-tot,K-S (sigma), and the maximal Galois extension of K in K-tot,K-S(sigma) by K-tot,K-S [sigma]. We prove that for almost all sigma is an element of Gal(K)(e) the absolute Galois group of K-tot,K-S [sigma] is isomorphic to the free product of (F) over cap (omega) and a free product of local factors over S.